Author(s):

Erdős, László

Title: 
Random matrices, loggases and Hölder regularity

Title Series: 
Proceedings of the International Congress of Mathematicians

Affiliation 
IST Austria 
Abstract: 
The WignerDysonGaudinMehta conjecture asserts that the local eigenvalue statistics of large real and complex Hermitian matrices with independent, identically distributed entries are universal in a sense that they depend only on the symmetry class of the matrix and otherwise are independent of the details of the distribution. We present the recent solution to this halfcentury old conjecture. We explain how stochastic tools, such as the Dyson Brownian motion, and PDE ideas, such as De GiorgiNashMoser regularity theory, were combined in the solution. We also show related results for loggases that represent a universal model for strongly correlated systems. Finally, in the spirit of Wigner’s original vision, we discuss the extensions of these universality results to more realistic physical systems such as random band matrices.

Keywords: 
De GiorgiNashMoser parabolic regularity, WignerDysonGaudinMehta universality, Dyson Brownian motion

Conference Title:

ICM: International Congress of Mathematicians

Volume: 
3

Conference Dates:

August 13  21, 2014

Conference Location:

Seoul, Korea

ISBN:

9788961058063

Publisher:

Kyung Moon SA Co. Ltd.

Location:

Seoul

Date Published:

20140801

Start Page: 
214

End Page:

236

Sponsor: 
The author is partially supported by SFBTR 12 Grant of the German Research Council and by ERC Advanced Grant, RANMAT 338804.

URL: 

Open access: 
yes (repository) 